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  • Produit eulérien motivique et courbes rationnelles sur les variétés toriques

  • Part of
  • David Bourqui
  • Journal:
    Compositio Mathematica / Volume 145 / Issue 6 / November 2009
    Published online by Cambridge University Press:
    03 December 2009, pp. 1360-1400
    Print publication:
    November 2009
    • Article
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  • We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.